Subgroups — #20272635

Order 6784.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810399	C₂	1	1	yes	maximal, characteristic
2	#16810344	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#346	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719908	C₄	1	2	yes	—
5	C₄×#88322	C₄	1	2	yes	—
6	C₂×C₁₆×C₅₃	C₄	1	2	yes	—
7	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
10	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
11	C₂×#88322	—	2	2	no	—
12	C₁₆×C₅₃	C₈	1	4	yes	complemented
13	#11719908	—	2	2	no	—
14	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
15	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
16	C₈×C₅₃	C₂×C₈	1	2	yes	—
17	C₈×C₅₃	M₁₆	1	2	yes	—
18	#88322	—	4	2	no	—
19	C₂²×C₅₃	#97	1	1	yes	characteristic
20	C₄×C₅₃	—	2	2	no	—
21	C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
22	C₄×C₅₃	#97	1	1	yes	characteristic
23	#1313	—	53	1	no	maximal, Sylow
24	C₂×C₅₃	#322	1	1	yes	characteristic
25	C₂×C₅₃	#347	1	2	yes	—
26	#346	—	53	1	no	—
27	#346	D₅₃	1	1	yes	characteristic
28	#322	—	53	1	no	—
29	C₅₃	#1313	1	1	yes	characteristic, complemented, Sylow
30	C₂×C₁₆	—	53	2	no	—
31	C₂×C₁₆	Dic₅₃	1	2	yes	—
32	C₄×C₈	—	53	2	no	—
33	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
34	C₂×C₈	—	106	2	no	—
35	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
36	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
37	C₄²	C₄×D₅₃	1	1	yes	characteristic
38	C₁₆	—	106	2	no	—
39	C₁₆	#88322	1	4	yes	complemented
40	C₂×C₄	#11719912	1	2	yes	—
41	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
42	C₈	—	212	2	no	—
43	C₈	#11719912	1	2	yes	—
44	C₈	C₂×#88322	1	2	yes	—
45	C₂²	#13133127	1	1	yes	characteristic
46	C₄	—	2	2	no	—
47	C₄	C₄×#88322	1	1	yes	characteristic
48	C₄	#13133127	1	1	yes	characteristic
49	C₂	#16810344	1	1	yes	characteristic
50	C₂	#16810400	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄×Dic₅₃
commutator	C₄×C₅₃	C₄×C₈
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#346	C₂
socle	C₂²×C₅₃	#97
hypercenter	#346	D₅₃
hypocenter	C₅₃	#1313
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #346

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
12. C₁₆×C₅₃	C₈	semidirect
29. C₅₃	#1313	semidirect
39. C₁₆	#88322	semidirect